Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey

Previous Article
Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise
DCDS-B Home
This Issue
Next Article
Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices

June 2018, 23(4): 1721-1737. doi: 10.3934/dcdsb.2018073

Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey

Na Min and Mingxin Wang ^,

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Mingxin Wang

Received August 2017 Published January 2018

Fund Project: This work was supported by NSFC Grant 11371113

In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

Keywords: Ratio-dependent prey-predator model, reaction-diffusion, Allee effect, protection zone, bifurcation.

Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 35B32, 35B35, 35B40, 35K57.

Citation: Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073

References:

[1]	W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949.Google Scholar
[2]	H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004. Google Scholar
[3]	R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. doi: 10.2307/1940007. Google Scholar
[4]	R. G. Casten and C. G. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Equat., 27 (1978), 266-273. doi: 10.1016/0022-0396(78)90033-5. Google Scholar
[5]	R. H. Cui, J. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129. doi: 10.1016/j.jde.2013.08.015. Google Scholar
[6]	Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Equat., 244 (2008), 61-86. doi: 10.1016/j.jde.2007.10.005. Google Scholar
[7]	Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Equat., 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007. Google Scholar
[8]	Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Equat., 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013. Google Scholar
[9]	S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10. doi: 10.1016/0025-5564(78)90025-1. Google Scholar
[10]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. Google Scholar
[11]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Ent., 91 (1959), 385-398. doi: 10.4039/Ent91385-7. Google Scholar
[12]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966. Google Scholar
[13]	P. E. Kloeden and C. Pötzsche, Dynamics of modified predator-prey models, Int. J. Bifurc. Chaos, 20 (2010), 2657-2669. doi: 10.1142/S0218127410027271. Google Scholar
[14]	Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634. doi: 10.1360/N012015-00233. Google Scholar
[15]	Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772. doi: 10.1007/s11784-016-0372-2. Google Scholar
[16]	Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal. RWA, 14 (2013), 1806-1816. doi: 10.1016/j.nonrwa.2012.11.012. Google Scholar
[17]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1973.Google Scholar
[18]	K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685. doi: 10.1090/S0002-9947-1995-1290727-7. Google Scholar
[19]	N. Min and M. X. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689. doi: 10.1016/j.camwa.2016.07.028. Google Scholar
[20]	L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/cln/006. Google Scholar
[21]	W.J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272. doi: 10.1016/j.jde.2016.06.022. Google Scholar
[22]	W. J. Ni and M. X. Wang, Dynamicl properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420. doi: 10.3934/dcdsb.2017172. Google Scholar
[23]	P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942. doi: 10.1017/S0308210500002742. Google Scholar
[24]	F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123-144. doi: 10.1016/j.ecocom.2016.07.001. Google Scholar
[25]	M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Phys. D, 196 (2004), 172-192. doi: 10.1016/j.physd.2004.05.007. Google Scholar
[26]	M. X. Wang, Nonlinear Elliptic Ppartial Differential Equations (in Chinese), Science Press, Beijing, 2010.Google Scholar
[27]	Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245. doi: 10.1016/j.nonrwa.2012.06.001. Google Scholar
[28]	J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004. Google Scholar
[29]	Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, The Introduction of Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 2011.Google Scholar
[30]	F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar
[31]	J. Zhou and C. L. Mu, Coexistence of a diffusive predator-prey model with Holling type-Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927. doi: 10.1016/j.jmaa.2011.07.027. Google Scholar

show all references

References:

[1]	W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949.Google Scholar
[2]	H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004. Google Scholar
[3]	R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. doi: 10.2307/1940007. Google Scholar
[4]	R. G. Casten and C. G. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Equat., 27 (1978), 266-273. doi: 10.1016/0022-0396(78)90033-5. Google Scholar
[5]	R. H. Cui, J. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129. doi: 10.1016/j.jde.2013.08.015. Google Scholar
[6]	Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Equat., 244 (2008), 61-86. doi: 10.1016/j.jde.2007.10.005. Google Scholar
[7]	Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Equat., 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007. Google Scholar
[8]	Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Equat., 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013. Google Scholar
[9]	S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10. doi: 10.1016/0025-5564(78)90025-1. Google Scholar
[10]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. Google Scholar
[11]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Ent., 91 (1959), 385-398. doi: 10.4039/Ent91385-7. Google Scholar
[12]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966. Google Scholar
[13]	P. E. Kloeden and C. Pötzsche, Dynamics of modified predator-prey models, Int. J. Bifurc. Chaos, 20 (2010), 2657-2669. doi: 10.1142/S0218127410027271. Google Scholar
[14]	Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634. doi: 10.1360/N012015-00233. Google Scholar
[15]	Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772. doi: 10.1007/s11784-016-0372-2. Google Scholar
[16]	Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal. RWA, 14 (2013), 1806-1816. doi: 10.1016/j.nonrwa.2012.11.012. Google Scholar
[17]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1973.Google Scholar
[18]	K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685. doi: 10.1090/S0002-9947-1995-1290727-7. Google Scholar
[19]	N. Min and M. X. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689. doi: 10.1016/j.camwa.2016.07.028. Google Scholar
[20]	L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/cln/006. Google Scholar
[21]	W.J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272. doi: 10.1016/j.jde.2016.06.022. Google Scholar
[22]	W. J. Ni and M. X. Wang, Dynamicl properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420. doi: 10.3934/dcdsb.2017172. Google Scholar
[23]	P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942. doi: 10.1017/S0308210500002742. Google Scholar
[24]	F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123-144. doi: 10.1016/j.ecocom.2016.07.001. Google Scholar
[25]	M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Phys. D, 196 (2004), 172-192. doi: 10.1016/j.physd.2004.05.007. Google Scholar
[26]	M. X. Wang, Nonlinear Elliptic Ppartial Differential Equations (in Chinese), Science Press, Beijing, 2010.Google Scholar
[27]	Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245. doi: 10.1016/j.nonrwa.2012.06.001. Google Scholar
[28]	J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004. Google Scholar
[29]	Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, The Introduction of Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 2011.Google Scholar
[30]	F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar
[31]	J. Zhou and C. L. Mu, Coexistence of a diffusive predator-prey model with Holling type-Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927. doi: 10.1016/j.jmaa.2011.07.027. Google Scholar

[1]	Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063
[2]	Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283
[3]	Renhao Cui, Haomiao Li, Linfeng Mei, Junping Shi. Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 791-807. doi: 10.3934/dcdsb.2017039
[4]	Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[5]	Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747
[6]	Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597
[7]	Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172
[8]	Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 303-315. doi: 10.3934/dcdss.2008.1.303
[9]	Inkyung Ahn, Wonlyul Ko, Kimun Ryu. Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey. Conference Publications, 2013, 2013 (special) : 11-19. doi: 10.3934/proc.2013.2013.11
[10]	Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[11]	Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[12]	Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423
[13]	Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[14]	Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321
[15]	Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[16]	Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040
[17]	Jaume Llibre, Claudio Vidal. Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1859-1867. doi: 10.3934/dcdsb.2016026
[18]	Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737
[19]	Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345
[20]	Mostafa Bendahmane. Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks & Heterogeneous Media, 2008, 3 (4) : 863-879. doi: 10.3934/nhm.2008.3.863

American Institute of Mathematical Sciences